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Mixed factorial
Mixed Factorial is a type of factorial coined by the googology wiki user, SpongeTechX. They use a number of mixed operations in order of growth rate. It always starts with addition, then multiplication, then exponentiation, etc. They can be iterated with a "*" 'symbol. An example of a mixed factorial is: *'mixed factorial of 3 (or 3*) = 1 + 2 x 3 If you want to go to exponentiation, or find out the mixed factorial of 4, you have to find 3 to the power of 4, which is 81. If you want to go even further, and find the mixed factorial of 5, take 81 and tetrate it by 5. Pentate the number you find (which is too large to write on this screen) by 6 to find the mixed factorial of 6, and so on. For this function to be well defined mixed factorials can only be written in one order, namely from addition to multiplication, then exponentiation etc. More examples to make all of this simpler are: *'4* = 1 + 2 x 3↑4' *'5* = 1 + 2 x 3↑4↑↑5' *'6* = 1 + 2 x 3↑4↑↑5↑↑↑6' *'7* = 1 + 2 x 3↑4↑↑5↑↑↑6↑↑↑↑7' Extended Mixed Factorials By repeating the process of taking mixed factorial we get numbers of form (n*)*, ((n*)*)* etc. or, equivalently, n**, n***, n****... These can be also written as n*2, n*3, etc. First Extension For even farther extension, we can say n*x*x*x*...x*x, with k x*x*x'''s can be iterated as '''n*k,x. As you can see, x represents the mixed factorial numbers, and k represents the amount of them. Second Extension Even farther extension can represented in this diagram: As you can see, 2 arrays are represented by n*k,x:2, 3 arrays are represented by n*k,x:3, 4 arrays are represented by n*k,x:4, and so on. Third Extension Farther extensions can be represented in the examples below: *'n*[k,x:(n*k,x)] = n*k,x#2' *'n*[k,x:(n*[k,x:(n*k,x)])] = n*k,x#3' *'n*[k,x:(n*[k,x:(n*[k,x:(n*k,x)])])] = n*k,x#4' n*k,x#5, n*k,x#6, etc. If this isn't very clear to you, get a piece of paper and try working this out. It's a bit complicated, but makes huge numbers. Fourth Extension The fourth extension can be represented in the examples below. As said before, try working it out on paper if you don't get it. It has multiple #'s. *'n*[k,x#(n*k,x)] = n*k,x##2 or n*2' *'n*[k,x#(n*[k,x#(n*k,x)])] = n*k,x##3 or n*3' *'n*[k,x#(n*[k,x#(n*[k,x#(n*k,x)])])] = n*k,x##4 or n*4' n*k,x##5,' n*k,x##6',' n*k,x##7',' '''etc. For three #'s: *'n*[k,x##(n*k,x)] = n*k,x###2' or '''n*2' *'n*[k,x##(n*[k,x##(n*k,x)])] = n*k,x###3' or n*3 Four #'s: *'n*[k,x###(n*k,x)] = n*k,x####2' or n*2 *'n*[k,x###(n*[k,x###(n*k,x)])] = n*k,x####3' or n*3 Five #'s: *'n*[k,x####(n*k,x)] = n*k,x#####2' or n*2 *'n*[k,x####(n*[k,x####(n*k,x)])] = n*k,x#####3' or n*3 Six #'s, seven #'s, and so on. Note: As for the | symbol, it is just a seperator. If it wasn't there, it would look like pentation instead of the amount of #'s. Fifth Extension *'n*[k,x#(n*n)|n] = n*2 or n*2' *'n*[k,x#(n*[k,x#(n*n)|n])|y] = n*3 or n*3' We have 2 ≍'s, 3 ≍'s, etc. On 3 in 2nd entry. *'n*[k,x≍(n*n)|n] = n*2' *'n*[k,x≍(n*[k,x≍(n*n)|n])|n] = n*3' On 4 in 2nd entry. *'n*[k,x#(n*n),3|n] = n*2' *'n*[k,x#(n*[k,x≍(n*n),3|n]),3|n] = n*3' We extend to 3 entries: *'n*[k,x#1,(n*n)|n] = n*2' *'n*[k,x#1,(n*[k,x#1,(n*n)|n])|y] = n*3' *'n*[k,x#1,(n*n),2|n] = n*2' *'n*[k,x#1,(n*[k,x#1,(n*n),2|n]),2|y] = n*3' And the four entries, five entries, etc. Sixth Extension Iterating large amounts of entries can be explained in the following: Let's use q''' as the amount of entries, and '''s as the number in the entries, we could iterate that as n*. . . s,s,s,s|3 (with q''' entries) as '''n*[k,x#s2q|3]. We defined n*[k,x#s21,2|3] = n*[k,x#s2n*[k,x#s2n*3|3]|3] and n*[k,x#s2q2q2|3] = n*[k,x#s2q,q...q,q|3] (q2 q's) Also n*[k,x#sw+1q|3] = n*[k,x#sws...sws|3] (q s's). Also we have nested arrays. For example: n*[k,x#s1,2q|3] = n*[k,x#s[n*[k,x#s[n*3]q|3]]q|3]. Seventh Extension *'n*[k,x#nn,2m-1n,2m-1|n] = n*2' *'n*[k,x#n[nn,2m-1n,2m-1]n,2m-1|n] = n*3' Other arrays work same. *'n*[k,x#nn,om-1o-1n,om-1|n] = n*2' *'n*[k,x#n[nn,om-1o-1n,om-1]o-1n,om-1|n] = n*3' Sources *SpongeTechX's Googology World See Also Category:Factorials Category:SpongeTechX Category:Functions